LAUSR

R f(x− C(s)) ds s , x ∈ R with an appropriate curve C(s) = (s, c(s)) in Rn. It is a very well-known fact due to E. M. Stein and S. Wainger [5] that the Hilbert transform along curves is bounded operator on Lp with 1 < p < ∞, when one takes the well-curved C as an appropriate curve. That is, one chooses C(s) = (s, c(s)) with a smooth c in Rn−1, and c(0) = 0 so that 〈{dkC(s) dsk ∣∣ s=0 } k=1,2,3,... 〉 = Rn. One easily notices that above operator does not contain any oscillating terms. It is S. Chandarana [1] who first tried to control this operator with an additional oscillating term, e −2πi|s|−β |s|α , since with the singular term such as 1 |s|α the operator is not bounded on L2. In his paper, the hypersingular integral is defined as